Passive tracer in non-Markovian, Gaussian velocity field

We consider the trajectory of a tracer that is the solution of an ordinary differential equation Ẋ(t)=V(t,X(t)),X(0)=0, with the right hand side, that is a stationary, zero-mean, Gaussian vector field with incompressible realizations. It is known, see Fannjiang and Komorowski (1999), Carmona and Xu (1996) and Komorowski et al. (2012), that X(t)∕t converges in law, as t→+∞, to a normal, zero mean vector, provided that the field V(t,x) is Markovian and has the spectral gap property. We wish to extend this result to the case when the field is not Markovian and its covariance matrix is given by a completely monotone Bernstein function.